#7562680

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 ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2302 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)(8,9)(10,12)(13,14,17,19)(15,18,22,20)(16,24,26,21)(23,25,28,27);(2,3)(6,9)(10,11)(14,19)(16,25)(18,20)(21,28)(23,24)(26,27);(1,2,4,3)(13,15,17,22)(14,20,19,18)(16,23,26,28)(21,25,24,27);(1,3,4,2)(13,16,17,26)(14,21,19,24)(15,23,22,28)(18,27,20,25);(5,7,8,12)(6,10,9,11)(14,19)(15,22)(21,24)(23,28)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72263913, #72263942, #72264000
maximal normal subgroup of	#72263913, #72263942, #72264000
maximal quotient of	#72264422

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469876);