#232057

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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

645 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)(17,18,21,25,22,26,27,24)(19,23,28,31,29,32,20,30);(1,2)(3,7)(4,8)(5,6)(9,11,13,16)(10,14,15,12)(17,19,27,20,22,29,21,28)(18,23,24,30,26,32,25,31);(1,3,5,4)(2,7,6,8)(9,12,16,10,13,14,11,15)(17,20,27,29,22,28,21,19)(18,23,25,31,26,32,24,30);(1,4,5,3)(2,7,6,8)(9,11,13,16)(10,14,15,12)(17,21,22,27)(18,24,26,25)(19,28,29,20)(23,30,32,31)

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,d^a=di,f^a=fh,c^b=ch,d^b=de,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343763, #72343872, #72343874
maximal normal subgroup of	#72343763, #72343872, #72343874

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139253);