#7065181

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1955 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)(13,14,20,28,19,23,18,17)(15,24,26,21,22,27,25,16);(5,6)(7,11)(8,10)(9,12)(13,15,19,22)(14,21,23,16)(17,27,28,24)(18,26,20,25);(2,4)(5,7,8,12)(6,9,10,11)(13,16,19,21)(14,22,23,15)(17,26,28,25)(18,27,20,24);(7,12)(9,11)(13,17,20,14,19,28,18,23)(15,16,26,24,22,21,25,27);(6,10)(9,11)(14,23)(16,21)(17,28)(24,27)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=hi,d^2=fi,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,e^d=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, 6972377);