#230879

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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

677 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)(10,14)(11,12)(15,16)(17,18,23,25,21,20,22,30)(19,26,29,32,27,24,28,31);(1,2,5,6)(3,7,4,8)(10,15)(14,16)(17,19,22,28,21,27,23,29)(18,24,30,32,20,26,25,31);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,16)(13,15)(17,20,21,18)(19,24,27,26)(22,25,23,30)(28,32,29,31);(1,4,5,3)(2,7,6,8)(9,11,13,12)(10,14,15,16)(18,20)(19,27)(25,30)(28,29)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,c^b=ch,d^b=de,d^c=di,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343540, #72343874, #72343884
maximal normal subgroup of	#72343540, #72343874, #72343884

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138075);