#7454367

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(33,34)(35,43)(36,41)(37,39)(38,47)(40,44)(42,46)(45,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,18)(19,25)(20,28)(21,23)(22,26)(24,29)(27,30)(31,32)(34,39)(35,36)(42,47)(44,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,22)(20,27)(21,26)(23,25)(24,30)(28,31)(29,32)(33,35)(34,40)(36,37)(38,45)(39,48)(41,42)(43,47)(44,46);(3,11)(6,9)(10,16)(13,15)(18,23)(19,26)(24,28)(27,32)(33,36,37,35)(34,41,39,43)(38,45,46,44)(40,47,48,42);(2,8)(6,9)(7,14)(13,15)(17,20,21,29)(18,24,23,28)(19,27,26,32)(22,30,25,31)(34,39)(35,36)(42,47)(44,45)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7361563);