#492832

order	512
nilpotency class	2
isoclinism	isoclinic class of #20338
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₄²
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

954 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224826, #72224874, #72224893, …
maximal normal subgroup of	#72224826, #72224874, #72224893, …
maximal quotient of	#72224814, #72227060, #72319129, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400028);