#6388495

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×#3115	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=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, 6295691);