#354102

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	C₂⁵	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2954 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)(7,11,17,23)(8,12)(9,13)(10,14)(15,27,29,21)(16,22,30,33)(18,24)(19,25)(20,26)(28,35,36,32)(31,34);(5,7)(6,11,14,23)(8,15)(9,16)(10,17)(12,21,25,27)(13,22,26,33)(18,28)(19,29)(20,30)(24,32,34,35)(31,36);(1,2,3,4)(5,8)(6,12)(7,15)(9,18)(10,19)(11,21)(13,24)(14,25)(16,28)(17,29)(20,31)(22,32)(23,27)(26,34)(30,36)(33,35);(5,9,10,20)(6,13,14,26)(7,16,17,30)(8,18,19,31)(11,22,23,33)(12,24,25,34)(15,28,29,36)(21,32,27,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^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, #72239777, #72239913, …
maximal normal subgroup of	#72239305, #72239777, #72239913, …
maximal quotient of	#72239189, #72239191, #72239587, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261298);