#7561652

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514196
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

2390 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283111, #72283701, #72283867
maximal normal subgroup of	#72283111, #72283701, #72283867

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7468848);