#493078

order	512
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=h,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,d^c=di,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226790, #72226795, #72226798, …
maximal normal subgroup of	#72226790, #72226795, #72226798, …
maximal quotient of	#72309743, #72309775, #72309843, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400274);