#6400467

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×#553	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1870 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)(5,6,9,12)(7,11,8,10)(13,14)(15,23)(16,22)(17,21)(18,20)(19,24)(25,28)(26,27);(5,6)(7,10)(8,11)(9,12)(13,15,17,24)(14,19,21,23)(16,26,18,25)(20,27,22,28);(5,7,9,8)(6,10,12,11)(14,20)(16,18)(19,27)(21,22)(23,28)(25,26);(5,7,9,8)(6,11,12,10)(14,21)(15,24)(20,22)(25,26);(1,3,4,2)(5,8,9,7)(6,11,12,10)(13,16,17,18)(14,20,21,22)(15,25,24,26)(19,28,23,27)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=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, 6307663);