#7519978

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	#45101	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7427174);