#6440569

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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
prime radical	#34632	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2991 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)(7,9)(8,10)(11,12)(13,14,16,21,17,20,24,18)(15,19,23,27,22,26,28,25);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(2,4)(6,10)(9,12)(13,15)(14,19)(16,23)(17,22)(18,25)(20,26)(21,27)(24,28);(2,4)(15,22)(19,26)(23,28)(25,27);(14,20)(18,21)(19,26)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282358, #72282497, #72282534
maximal normal subgroup of	#72282358, #72282497, #72282534

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347765);