#493685

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1186 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)(5,6,10,16,9,17,26,36,11,15,28,30,27,14,19,35)(7,18,22,13,21,12,29,32,20,33,8,23,25,31,24,34);(1,2,3,4)(5,7,9,21,11,20,27,25)(6,12,14,18,15,31,17,33)(8,19,24,10,22,26,29,28)(13,30,34,36,23,16,32,35);(5,8)(6,13)(7,19)(9,24)(10,21)(11,22)(12,30)(14,34)(15,23)(16,31)(17,32)(18,36)(20,26)(25,28)(27,29)(33,35);(7,20)(8,22)(12,31)(13,23)(18,33)(21,25)(24,29)(32,34)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239244, #72239254, #72239289, …
maximal normal subgroup of	#72239244, #72239254, #72239289, …
maximal quotient of	#72239339

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400881);