#7592906

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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×#121 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=g,d^2=gi,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^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285178, #72285302
maximal normal subgroup of	#72285178, #72285302

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7500102);