#6393425

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,22,19,24,17,23)(15,20,26,36,27,35,28,34)(16,29,30,21,32,37,31,38)(25,39,42,33,41,43,40,44);(5,7)(6,9)(8,12)(10,11)(13,15,19,27)(14,20,24,35)(16,25,32,41)(17,28,18,26)(21,33,38,44)(22,36,23,34)(29,39,37,43)(30,42,31,40);(1,2,3,4)(13,16)(14,21)(15,25)(17,30)(18,31)(19,32)(20,33)(22,29)(23,37)(24,38)(26,40)(27,41)(28,42)(34,43)(35,44)(36,39);(7,12)(9,11)(13,17,19,18)(14,22,24,23)(15,26,27,28)(16,30,32,31)(20,34,35,36)(21,29,38,37)(25,40,41,42)(33,43,44,39);(6,10)(9,11)(15,27)(20,35)(25,41)(26,28)(33,44)(34,36)(39,43)(40,42)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=dfi,e^b=ei,g^c=gi,h^c=hi

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, 6300621);