#227622

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
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

623 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,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,d^c=dh,f^c=fi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319585, #72322005, #72322149
maximal normal subgroup of	#72319585, #72322005, #72322149

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134818);