#364425

order	512
nilpotency class	3
isoclinism	isoclinic class of #196290
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2434 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)(5,6,10,15,11,16,9,14)(7,12,18,27,20,28,19,32)(8,21,24,13,25,30,23,31)(17,29,33,35,34,36,22,26);(2,4)(5,7,10,18,11,20,9,19)(6,12,15,27,16,28,14,32)(8,22,24,17,25,33,23,34)(13,29,30,35,31,36,21,26);(5,8,11,25)(6,13,16,31)(7,17,20,34)(9,24,10,23)(12,26,28,35)(14,30,15,21)(18,22,19,33)(27,36,32,29);(2,4)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285452, #72285470, #72285641
maximal normal subgroup of	#72285452, #72285470, #72285641
maximal quotient of	#72316755, #72317024

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271621);