#6377737

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×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	C₂³×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2024 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)(7,11)(8,10)(9,12)(13,14,17,22)(15,24,25,19)(16,28,29,21)(18,20,31,23)(26,33,38,37)(27,32,39,34)(30,36,41,35)(40,43,44,42);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,26)(17,25)(18,27)(20,34)(21,33)(22,24)(23,32)(28,37)(29,38)(30,40)(31,39)(35,43)(36,42)(41,44);(7,12)(9,11)(14,20)(15,25)(18,31)(19,32)(21,35)(22,23)(24,34)(26,38)(28,36)(30,41)(33,42)(37,43);(6,10)(9,11)(13,16)(14,21)(15,26)(17,29)(18,30)(19,33)(20,35)(22,28)(23,36)(24,37)(25,38)(27,40)(31,41)(32,42)(34,43)(39,44);(1,3,4,2)(15,25)(16,29)(19,24)(21,28)(27,39)(30,41)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^b=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, 6284933);