#7523200

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1920 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,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=cg,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

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263905, #72263916, #72263968
maximal normal subgroup of	#72263905, #72263916, #72263968
maximal quotient of	#72264389, #72265020

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7430396);