#359604

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3844 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)(7,17,21,12)(8,14,9,23)(10,16)(11,15)(13,22,31,24)(18,34)(19,28)(20,33,25,29)(26,32)(27,35)(30,36);(2,4)(5,7)(6,12)(8,18)(9,19)(10,25)(11,21)(13,30)(14,28)(15,17)(16,33)(20,26)(22,36)(23,34)(24,35)(27,31)(29,32);(5,8,11,9)(6,13,15,31)(7,18,21,19)(10,24,26,22)(12,27,17,30)(14,16,23,32)(20,36,25,35)(28,29,34,33);(2,4)(5,9,11,8)(6,14,15,23)(7,19,21,18)(10,24,26,22)(12,28,17,34)(13,16,31,32)(20,36,25,35)(27,29,30,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303071, #72303077, #72303269, …
maximal normal subgroup of	#72303071, #72303077, #72303269, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266800);