#7221098

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	#31001	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 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)(11,15)(14,16)(17,18,21,26)(19,28,30,23)(20,25,33,41)(22,24,35,27)(29,40,32,37)(31,38,44,36)(34,39,46,42)(43,48,45,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,13)(14,15)(17,19,21,30)(18,23,26,28)(20,32,33,29)(22,31,35,44)(24,36,27,38)(25,40,41,37)(34,45,46,43)(39,48,42,47);(10,13)(14,16)(18,24)(22,35)(23,36)(25,39)(26,27)(28,38)(31,44)(34,46)(37,47)(40,48)(41,42)(43,45);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,39)(26,41)(27,42)(28,40)(30,32)(31,43)(35,46)(36,47)(38,48)(44,45);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,26)(20,33)(23,28)(24,27)(29,32)(34,46)(36,38)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7128294);