#370000

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1020 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)(5,6,10,16,11,17,9,15)(7,12,21,28,19,14,20,27)(8,13,24,32,23,31,25,33)(18,30,22,36,34,26,35,29);(1,3,4,2)(5,7)(6,12)(8,22,23,35)(9,20)(10,21)(11,19)(13,29,31,36)(14,17)(15,27)(16,28)(18,24,34,25)(26,32,30,33);(5,8,9,25,11,23,10,24)(6,13,15,33,17,31,16,32)(7,18,20,35,19,34,21,22)(12,26,27,36,14,30,28,29);(5,7)(6,14)(8,18)(9,20)(10,21)(11,19)(12,17)(13,30)(15,28)(16,27)(22,24)(23,34)(25,35)(26,31)(29,33)(32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265431, #72307320, #72307350
maximal normal subgroup of	#72265431, #72307320, #72307350

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277196);