#7064758

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
prime radical	C₂×#3308	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1913 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)(13,14,19,20,18,23,17,22)(15,24,26,27,25,16,21,28);(5,6)(7,11)(8,10)(9,12)(14,20)(15,25)(16,27)(17,19)(22,23)(24,28);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,16)(17,21)(18,25)(19,26)(20,28)(22,27)(23,24);(7,12)(9,11)(13,16,19,28,18,24,17,27)(14,21,20,15,23,26,22,25);(6,10)(9,11)(13,17,18,19)(14,20,23,22)(15,26,25,21)(16,27,24,28)

Polycyclic presentation:

2^9|a^2=fg,c^2=hi,d^2=fi,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, 6971954);