#354829

order	512
nilpotency class	3
isoclinism	isoclinic class of #453
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3502 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)(7,16)(8,13)(9,23)(10,14)(11,15)(12,18)(17,30)(19,27)(20,33)(21,28)(22,29)(24,26)(25,31)(32,36)(34,35);(5,7,10,18,11,19,9,24)(6,12,14,27,15,26,23,16)(8,17,21,31,22,32,20,34)(13,25,28,36,29,35,33,30);(5,6)(7,12)(8,13)(9,23)(10,14)(11,15)(16,24)(17,25)(18,27)(19,26)(20,33)(21,28)(22,29)(30,34)(31,36)(32,35);(5,8,10,21,11,22,9,20)(6,13,14,28,15,29,23,33)(7,17,18,31,19,32,24,34)(12,25,27,36,26,35,16,30)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239887, #72257236, #72257245
maximal normal subgroup of	#72239887, #72257236, #72257245
maximal quotient of	#72239890, #72317647

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262025);