#482994

order	512
nilpotency class	4
isoclinism	isoclinic class of #482956
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#62363	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

698 conjugacy classes of subgroups; only the 236 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 236 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=gh,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=dghi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390190);