#363501

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1262 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^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	#72238782, #72238790, #72239720, …
maximal normal subgroup of	#72238782, #72238790, #72239720, …
maximal quotient of	#72237297

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270697);