#494638

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1368 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)(5,6,9,14,11,15,10,25)(7,16,19,32,17,30,20,35)(8,21,22,27,24,36,23,13)(12,26,29,33,28,34,31,18);(5,7,8,18,9,19,22,26,11,17,24,33,10,20,23,34)(6,12,27,30,25,31,21,32,15,28,13,16,14,29,36,35);(2,4);(7,17)(12,28)(16,30)(18,33)(19,20)(26,34)(29,31)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249892, #72249906, #72249915, …
maximal normal subgroup of	#72249892, #72249906, #72249915, …
maximal quotient of	#72265732

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401834);