#365079

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	#2535
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1622 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280138, #72280200, #72280229
maximal normal subgroup of	#72280138, #72280200, #72280229

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272275);