#6401019

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1767 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^c=ei,g^c=gi,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, 6308215);