#6458511

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2076 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,5)(3,7,8,6)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,31)(21,26)(22,29)(23,27)(24,25)(30,32);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,31,23,30,22,25)(18,24,29,32,27,20,26,28);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(17,20)(18,25)(19,29)(21,28)(22,32)(23,24)(26,30)(27,31);(11,15)(13,16)(19,30)(20,24)(25,31)(28,32);(17,21,23,22)(18,26,27,29)(19,25,30,31)(20,28,24,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281400, #72281454
maximal normal subgroup of	#72281400, #72281454

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365707);