#483475

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂×#41	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

415 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 242 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,20,25,23,27,22,37)(19,28,30,43,33,45,32,48)(21,34,29,42,36,39,35,26)(24,38,40,47,41,46,44,31);(17,19,29,38,22,32,21,31,23,33,35,46,20,30,36,47)(18,24,39,48,25,40,26,28,27,41,34,43,37,44,42,45);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,20,23,22)(18,25,27,37)(19,30,33,32)(21,29,36,35)(24,40,41,44)(26,34,42,39)(28,43,45,48)(31,38,47,46);(3,9)(6,12)(10,15)(13,16)(17,20,23,22)(18,25,27,37)(19,30,33,32)(21,29,36,35)(24,40,41,44)(26,34,42,39)(28,43,45,48)(31,38,47,46)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=i,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,d^c=dhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234338, #72234342, #72234349, …
maximal normal subgroup of	#72234338, #72234342, #72234349, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390671);