#208169

order	512
nilpotency class	3
isoclinism	isoclinic class of #162406
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#35278	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1082 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)(3,6)(4,5)(7,8)(9,10,11,14,12,15,16,13)(18,22)(20,23)(24,28)(25,29)(27,30)(31,32);(10,13)(11,16)(14,15)(17,18,23,29,21,25,20,22)(19,24,30,32,26,31,27,28);(2,5)(6,8)(17,19,21,26)(18,24,25,31)(20,27,23,30)(22,28,29,32);(1,3,4,7)(2,6,5,8)(19,26)(24,31)(27,30)(28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=h,d^2=f,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=dh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115365);