#7283752

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1656 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7190948);