#7006762

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2205 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)(13,14,19,23,20,24,18,27)(15,25,17,22,21,28,26,16);(5,6)(7,11)(8,10)(9,12)(13,15)(14,16)(17,18)(19,26)(20,21)(22,24)(23,28)(25,27);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,24)(17,27)(18,28)(19,25)(20,22)(23,26);(7,12)(9,11)(13,17,18,15,20,26,19,21)(14,22,27,25,24,16,23,28);(6,10)(9,11)(13,18,20,19)(14,23,24,27)(15,26,21,17)(16,25,22,28)

Polycyclic presentation:

2^9|a^2=fgi,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, 6913958);