#370144

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1285 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)(7,11)(8,19,22,16)(9,23)(10,18)(12,15)(13,21,29,27)(14,24)(17,25)(20,33,36,28)(26,30)(31,34,32,35);(2,4)(5,7,10,6,12,18,11,15)(8,19,21,29,22,16,27,13)(9,17,26,14,25,23,24,30)(20,32,35,33,36,31,34,28);(1,2,3,4)(5,8,12,22)(6,13,15,29)(7,16,18,19)(9,20,25,36)(10,21,11,27)(14,28,30,33)(17,31,23,32)(24,34,26,35);(5,9)(6,14)(7,17)(8,20)(10,26)(11,24)(12,25)(13,28)(15,30)(16,31)(18,23)(19,32)(21,35)(22,36)(27,34)(29,33)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277340);