#6467307

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2130 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)(7,10)(8,9)(11,12);(2,4)(13,14,19,21)(15,22,26,23)(16,18,27,17)(20,24,28,25);(6,9)(10,12)(13,15,18,25,19,26,17,24)(14,20,16,22,21,28,27,23);(13,16,18,21,19,27,17,14)(15,23,25,20,26,22,24,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,16,21,27)(15,24,26,25)(20,22,28,23)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249920, #72256956, #72256990, …
maximal normal subgroup of	#72249920, #72256956, #72256990, …
maximal quotient of	#72264296, #72264332, #72264355

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374503);