#500246

order	512
nilpotency class	4
isoclinism	isoclinic class of #499803
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,b^a=be,c^a=cf,d^a=dei,g^a=ghi,c^b=cg,d^b=di,f^b=fh,h^c=hi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317920, #72317941, #72317969, …
maximal normal subgroup of	#72317920, #72317941, #72317969, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407442);