#354105

order	512
nilpotency class	3
isoclinism	isoclinic class of #1913
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2842 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,e^b=ei,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239305
maximal normal subgroup of	#72239305
maximal quotient of	#72239287, #72239293, #72239384

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261301);