#6440098

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

Polycyclic presentation:

2^9|a^2=i,b^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,d^c=di,g^c=gi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307139, #72307285, #72307767
maximal normal subgroup of	#72307139, #72307285, #72307767

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347294);