#6352140

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352061
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,7)(3,5,8,6)(9,10,14,18)(11,15,21,27)(12,16,23,29)(13,17,24,30)(19,31,33,25)(20,26,34,37)(22,28,35,38)(32,39,40,36);(2,5)(3,8)(6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,25)(17,26)(18,27)(22,32)(23,33)(24,34)(28,36)(29,31)(30,37)(35,40)(38,39);(9,12,14,23)(10,16,18,29)(11,19,21,33)(13,22,24,35)(15,25,27,31)(17,28,30,38)(20,32,34,40)(26,36,37,39);(9,13)(10,17)(11,20)(12,22)(14,24)(15,26)(16,28)(18,30)(19,32)(21,34)(23,35)(25,36)(27,37)(29,38)(31,39)(33,40);(13,24)(17,30)(20,34)(22,35)(26,37)(28,38)(32,40)(36,39)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281484, #72281528, #72281671
maximal normal subgroup of	#72281484, #72281528, #72281671

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259336);