#7142953

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45134	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2025 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050149);