#7503168

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2766 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)(15,24)(16,29)(17,33)(18,23)(19,22)(20,27)(21,32)(25,35)(26,36)(28,40)(30,37)(31,38)(34,39)(41,42)(43,44);(5,7)(6,9)(8,12)(10,11)(13,15,18,27)(14,20,23,24)(16,30,32,43)(17,28,19,26)(21,37,29,44)(22,40,33,36)(25,34,42,31)(35,38,41,39);(2,4)(13,16,18,32)(14,21,23,29)(15,25,27,42)(17,31,19,34)(20,35,24,41)(22,38,33,39)(26,30,28,43)(36,44,40,37);(2,4)(6,10)(7,12)(13,14)(15,24)(16,21)(17,33)(18,23)(19,22)(20,27)(25,41)(26,36)(28,40)(29,32)(30,44)(31,39)(34,38)(35,42)(37,43);(13,17,18,19)(14,22,23,33)(15,26,27,28)(16,31,32,34)(20,36,24,40)(21,38,29,39)(25,30,42,43)(35,44,41,37)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410364);