#7006752

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=f,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, 6913948);