#370157

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1287 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:

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

Polycyclic presentation:

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

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, 277353);