#207123

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1006 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,26,21,24,20,25)(19,23,28,30,27,31,29,32);(2,5)(6,7)(9,11,15,10,13,16,12,14)(17,19,20,28,21,27,22,29)(18,23,26,32,24,31,25,30);(1,3,4,8)(2,6,5,7)(10,14)(11,16)(18,24)(19,27)(25,26)(28,29);(17,20,21,22)(18,25,24,26)(19,28,27,29)(23,30,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fh,d^2=i,e^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ch,e^b=ei,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114319);