#7007129

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2162 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)(13,14,18,22,19,21,27,23)(15,16,17,26,24,25,28,20);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,23)(17,27)(18,28)(19,24)(21,26)(22,25);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,17)(15,23)(18,26)(19,25)(20,27)(21,28)(22,24);(7,12)(9,11)(13,17,27,15,19,28,18,24)(14,20,23,25,21,26,22,16);(6,10)(9,11)(14,21)(16,25)(20,26)(22,23)

Polycyclic presentation:

2^9|a^2=fg,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6914325);