#7003201

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

2075 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)(13,14,18,16)(15,21,24,28)(17,26,22,20)(19,27,23,25);(5,6)(7,11)(8,10)(9,12)(13,15,22,27,18,24,17,25)(14,19,26,28,16,23,20,21);(2,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,23)(16,18)(17,20)(19,24)(21,27)(22,26)(25,28);(7,12)(9,11)(13,16,18,14)(15,21,24,28)(17,20,22,26)(19,27,23,25);(6,10)(9,11)(14,16)(15,24)(20,26)(25,27)

Polycyclic presentation:

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

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, 6910397);