#7563640

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1771 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=gi,d^2=g,e^2=f,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=cg,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, 7470836);