#480768

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

840 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=chi,d^b=di,e^b=ei,f^b=fg,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387964);