#382452

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3634 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280665, #72306972, #72306998, …
maximal normal subgroup of	#72280665, #72306972, #72306998, …
maximal quotient of	#72308370

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289648);