#7054062

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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:

(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,16)(15,22,25,28)(17,21,19,26)(20,27,23,24);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,18)(17,26)(19,21)(22,24)(23,25)(27,28);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,23)(17,24)(18,25)(19,27)(21,28)(22,26);(2,4)(7,12)(9,11)(13,16,18,14)(15,23,25,20)(17,21,19,26)(22,24,28,27);(6,10)(9,11)(13,17,18,19)(14,21,16,26)(15,24,25,27)(20,28,23,22)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961258);