#354906

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262102);