#359747

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2296 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)(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,23)(19,27)(20,22)(24,28)(29,30)(31,32);(9,11,12,16)(10,13,14,15)(17,18)(19,28)(20,25)(21,23)(22,26)(24,29)(27,31)(30,32);(2,4)(17,19)(18,24)(20,29)(21,30)(22,27)(23,31)(25,32)(26,28);(10,14)(13,15)(17,20,21,22)(18,25,23,26)(19,29,30,27)(24,32,31,28)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240778, #72240780, #72240782
maximal normal subgroup of	#72240778, #72240780, #72240782

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266943);