#482258

order	512
nilpotency class	4
isoclinism	isoclinic class of #11753
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	#34963	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,f^c=fi,f^e=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281322, #72308506, #72308524, …
maximal normal subgroup of	#72281322, #72308506, #72308524, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389454);