#6454476

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2182 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)(15,23)(16,21)(17,19)(18,22)(20,26)(24,28)(25,27);(6,9)(10,12)(14,19)(16,17)(20,27)(21,22)(23,28)(24,26);(13,15,18,25)(14,20,22,28)(16,24,17,26)(19,27,21,23);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,21,22,19)(15,24,25,26)(20,23,28,27)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307108, #72307149, #72307412
maximal normal subgroup of	#72307108, #72307149, #72307412

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361672);