#361677

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×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	#2408
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3508 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)(5,6,10,14,11,15,26,32)(7,9,18,25,17,24,13,31)(8,12,23,30,21,28,22,29)(16,33,35,27,34,20,19,36);(2,4)(5,7,10,18,11,17,26,13)(6,9,14,25,15,24,32,31)(8,19,22,34,21,35,23,16)(12,27,29,33,28,36,30,20);(5,8)(6,12)(7,16)(9,20)(10,23)(11,21)(13,19)(14,30)(15,28)(17,34)(18,35)(22,26)(24,33)(25,36)(27,31)(29,32);(2,4)(5,9)(6,13)(7,14)(8,20)(10,25)(11,24)(12,19)(15,18)(16,30)(17,32)(21,33)(22,27)(23,36)(26,31)(28,35)(29,34)

Polycyclic presentation:

2^9|a^2=h,b^2=g,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	#72214895, #72237259, #72239707, …
maximal normal subgroup of	#72214895, #72237259, #72239707, …
maximal quotient of	#72237274, #72265739, #72280475, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268873);