#7149026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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 ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2241 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)(7,11)(10,12)(13,14,17,21,18,22,16,20)(15,19,25,26,23,28,24,27);(1,2,3,4)(6,9)(7,11)(13,15)(14,19)(16,24)(17,25)(18,23)(20,27)(21,26)(22,28);(5,6)(7,10)(8,9)(11,12)(14,20)(15,23)(16,17)(19,26)(21,22)(27,28);(2,4)(6,9)(10,12)(13,16,18,17)(14,20,22,21)(15,24,23,25)(19,27,28,26);(5,7,8,11)(6,10,9,12)(15,23)(19,28)(24,25)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056222);