#384344

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1326 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,26,28,31)(20,22,29,25)(24,27,30,32);(1,2)(3,4)(18,22)(19,27)(20,29)(23,25)(26,31)(28,32);(2,4)(9,11,12,16)(10,13,14,15)(18,23)(22,25)(24,30)(26,31);(10,14)(13,15)(17,19,20,27,21,28,29,32)(18,24,25,31,23,30,22,26)

Polycyclic presentation:

2^9|a^2=h,c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240709, #72240759, #72240765, …
maximal normal subgroup of	#72240709, #72240759, #72240765, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291540);