#493974

order	512
nilpotency class	4
isoclinism	isoclinic class of #480263
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311438, #72311460, #72311468, …
maximal normal subgroup of	#72311438, #72311460, #72311468, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401170);