#206971

order	512
nilpotency class	3
isoclinism	isoclinic class of #162118
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	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1384 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,13,15,16)(12,14)(18,22,23,24);(2,5)(6,7)(9,11,12,15)(10,13)(14,16)(17,18,20,24,21,23,19,22);(1,3,4,8)(2,6,5,7)(10,14)(13,16);(11,15)(13,16)(17,19,21,20)(18,22,23,24)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=fh,d^2=hi,e^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,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, 114167);