#359664

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62811	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3886 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:

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

Polycyclic presentation:

2^9|a^2=h,e^2=i,f^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,f^c=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317334, #72323003
maximal normal subgroup of	#72317334, #72323003

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266860);