#7006748

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2205 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913944);