#362627

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 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)(5,6,9,12,11,16,10,15)(7,13,17,25,19,27,24,29)(8,20,22,14,21,26,23,30)(18,32,31,28,34,35,33,36);(2,4)(6,12,16,15)(7,17,19,24)(8,21)(13,27)(14,20,30,26)(18,33,34,31)(22,23)(25,29);(2,4)(5,7)(6,13)(8,18)(9,17)(10,24)(11,19)(12,25)(14,28)(15,29)(16,27)(20,32)(21,34)(22,31)(23,33)(26,35)(30,36);(2,4)(5,8,9,22,11,21,10,23)(6,14,15,20,16,30,12,26)(7,18,17,31,19,34,24,33)(13,28,29,32,27,36,25,35)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269823);