#7611530

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2116 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7518726);