#227156

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	#18494	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

640 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dei,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287474, #72322026, #72322149
maximal normal subgroup of	#72287474, #72322026, #72322149

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134352);