#7527359

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518904
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

2375 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)(15,20)(16,25)(17,27)(18,22)(19,23)(21,26)(24,28);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,23,18,24,26,16)(14,19,21,20,22,25,27,28);(6,10)(7,11);(2,4)(6,10)(9,12);(13,16,26,24,18,23,17,15)(14,20,27,19,22,28,21,25)

Polycyclic presentation:

2^9|a^2=g,b^2=f,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307660, #72307669, #72307944
maximal normal subgroup of	#72307660, #72307669, #72307944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7434555);