#354038

order	512
nilpotency class	3
isoclinism	isoclinic class of #1909
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	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2882 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,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	#72252118, #72252121, #72252126, …
maximal normal subgroup of	#72252118, #72252121, #72252126, …
maximal quotient of	#72314612

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261234);