#6394641

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,g^c=gi,e^d=ei

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, 6301837);