#225779

order	512
nilpotency class	3
isoclinism	isoclinic class of #19755
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	D₄×C₄⋊C₄
prime radical	C₂×#135	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

530 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321745, #72322026, #72322038
maximal normal subgroup of	#72321745, #72322026, #72322038

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132975);