#353851

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

870 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240734, #72264451
maximal normal subgroup of	#72240734, #72264451

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261047);