#6448710

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3499 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305352, #72305359, #72305366
maximal normal subgroup of	#72305352, #72305359, #72305366
maximal quotient of	#72305738, #72305760

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355906);