#362633

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1344 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:

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

Polycyclic presentation:

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

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, 269829);