#377421

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1105 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=eh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284617);