#6440003

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2265 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307078, #72307149, #72307385
maximal normal subgroup of	#72307078, #72307149, #72307385

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347199);