#7169786

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994171
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
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1895 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7076982);