#6373752

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3153	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(2,4)(5,6,8,7)(9,10)(11,19)(12,17)(13,25)(14,16)(15,22)(18,26)(20,32)(21,31)(23,29)(24,37)(27,35)(28,33)(30,36)(34,39)(38,40);(9,11,13,22,14,21,26,35)(10,15,18,19,16,27,25,31)(12,20,24,33,23,34,36,40)(17,28,30,32,29,38,37,39);(1,2,3,4)(10,16)(15,27)(17,29)(18,25)(19,31)(28,38)(30,37)(32,39);(5,7,8,6)(9,12)(10,17)(11,20)(13,24)(14,23)(15,28)(16,29)(18,30)(19,32)(21,34)(22,33)(25,37)(26,36)(27,38)(31,39)(35,40);(11,21)(12,23)(15,27)(17,29)(19,31)(22,35)(24,36)(30,37)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gi,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,e^b=ei,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280948);