#7674632

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3489	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7816 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,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, 7581828);