#6439998

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

2289 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)(13,14,19,23)(15,20,16,25)(17,22,18,27)(21,24,28,26);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,19)(17,26)(18,24)(21,22)(23,25)(27,28);(2,4)(13,16,19,15)(14,21,23,28)(17,26,18,24)(20,27,25,22);(2,4)(13,17,19,18)(14,22,23,27)(15,24,16,26)(20,28,25,21);(5,7,8,11)(6,10,9,12)(15,16)(20,25)(21,28)(24,26)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=f,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	#72307132, #72307285, #72307385
maximal normal subgroup of	#72307132, #72307285, #72307385

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347194);