#7527572

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#120 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2105 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,e^2=fi,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^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, 7434768);