#7562207

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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×#120	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2228 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fg,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,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, 7469403);