#7151106

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
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1884 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)(13,14,20,15,19,17,18,22)(16,21,27,23,24,26,25,28);(6,9)(7,11)(13,15,19,22)(14,18,17,20)(16,23,24,28)(21,25,26,27);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,23)(17,26)(18,25)(19,24)(20,27)(22,28);(6,9)(10,12)(13,17,18,15,19,14,20,22)(16,21,25,28,24,26,27,23);(5,7,8,11)(6,10,9,12)(16,24)(21,26)(23,28)(25,27)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,d^2=f,e^2=h,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, 7058302);