#6454571

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437748
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3444	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3404 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)(15,22)(16,20)(17,26)(18,21)(19,24)(23,28)(25,27);(1,2)(3,4)(13,15,17,24,18,25,16,23)(14,19,20,22,21,28,26,27);(2,4)(6,9)(10,12)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,28,27);(2,4)(13,16,18,17)(14,20,21,26)(15,24,25,23)(19,27,28,22);(5,7,8,11)(6,10,9,12)(15,25)(19,28)(22,27)(23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283084
maximal normal subgroup of	#72283084

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361767);