#6376446

order	512
nilpotency class	3
isoclinism	isoclinic class of #60768
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

2486 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=fg,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^b=ei,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6283642);