#6667592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2915 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:

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

Polycyclic presentation:

2^9|a^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264133, #72264175, #72264241
maximal normal subgroup of	#72264133, #72264175, #72264241
maximal quotient of	#72265067

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6574788);