#7145808

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2269 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)(7,11)(10,12)(13,14,18,27,17,22,16,21)(15,20,24,19,26,23,25,28);(6,9)(7,11)(13,15)(14,19)(16,24)(17,26)(18,25)(20,27)(21,23)(22,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,26)(18,24)(19,27)(21,28)(22,23);(6,9)(10,12)(13,14,16,21,17,22,18,27)(15,23,25,19,26,20,24,28);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,21,22,27)(15,24,26,25)(19,23,28,20)

Polycyclic presentation:

2^9|a^2=fi,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,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, 7053004);