#493228

order	512
nilpotency class	2
isoclinism	isoclinic class of #180873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

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×D₄	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

707 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)(5,6,9,15,10,12,23,25,11,16,22,30,8,13,21,27)(7,14,19,29,17,24,31,34,20,28,33,36,18,26,32,35);(2,4)(6,12,16,13)(7,17,20,18)(9,22)(14,28)(15,27,30,25)(19,32,33,31)(21,23)(24,26);(6,13,16,12)(7,17,20,18)(9,22)(15,25,30,27)(19,32,33,31)(21,23)(29,36)(34,35);(5,7,9,19,10,17,23,31,11,20,22,33,8,18,21,32)(6,14,15,29,12,24,25,34,16,28,30,36,13,26,27,35)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239337, #72239417, #72239467, …
maximal normal subgroup of	#72239337, #72239417, #72239467, …
maximal quotient of	#72318580, #72318696, #72318755, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400424);