#7503142

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3012	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2826 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:

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

Polycyclic presentation:

2^9|a^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410338);