#7521992

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2203 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=f,e^2=fi,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263942, #72263962, #72263968
maximal normal subgroup of	#72263942, #72263962, #72263968
maximal quotient of	#72264394, #72265020

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7429188);