#7674638

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581834);