#6393447

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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×#3547	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3222 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6300643);