#7551438

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

2180 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)(13,14,17,21)(15,19,23,18)(16,20,24,27)(22,25,28,26);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,23)(19,21)(22,25)(24,27)(26,28);(6,9)(10,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,25)(24,28)(26,27);(5,7,8,12)(6,10,9,11)(13,16,17,24)(14,20,21,27)(15,22,23,28)(18,26,19,25);(1,3,4,2)(14,21)(15,23)(20,27)(22,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,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, #72263991, #72264032
maximal normal subgroup of	#72263913, #72263991, #72264032
maximal quotient of	#72264493

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7458634);