#492979

order	512
nilpotency class	2
isoclinism	isoclinic class of #180864
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1022 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)(5,6,9,16,10,12,23,25,11,14,22,29,8,15,21,30)(7,13,19,28,17,24,31,34,20,26,33,35,18,27,32,36);(2,4)(6,12,14,15)(7,17,20,18)(9,22)(13,26)(16,30,29,25)(19,32,33,31)(21,23)(24,27);(5,7,9,19,10,17,23,31,11,20,22,33,8,18,21,32)(6,13,16,28,12,24,25,34,14,26,29,35,15,27,30,36);(6,14)(12,15)(13,26)(16,29)(24,27)(25,30)(28,35)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239415, #72239417, #72239430, …
maximal normal subgroup of	#72239415, #72239417, #72239430, …
maximal quotient of	#72319160, #72319242, #72320092

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400175);