#7065140

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1897 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6972336);