#7542002

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1916 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7449198);