#495272

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1332 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226792, #72226795, #72226796, …
maximal normal subgroup of	#72226792, #72226795, #72226796, …
maximal quotient of	#72285524, #72319589, #72342591

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402468);