#232121

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

620 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=efh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=dei,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343763, #72343870, #72343877
maximal normal subgroup of	#72343763, #72343870, #72343877

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139317);