#7714882

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7858 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)(7,8)(9,10)(11,12)(17,29)(22,31)(25,36)(27,40)(33,41)(34,39)(38,43)(42,44);(1,2)(3,4)(14,19)(15,23)(21,26)(22,31)(24,35)(25,36)(34,39)(38,43);(6,10)(7,11)(13,14)(15,20)(16,26)(17,22)(18,19)(21,28)(23,30)(24,37)(25,33)(27,39)(29,31)(32,35)(34,40)(36,41)(38,44)(42,43);(5,7,9,11)(6,8,10,12)(13,15,18,23)(14,20,19,30)(16,24,28,35)(17,25,29,36)(21,32,26,37)(22,33,31,41)(27,38,40,43)(34,42,39,44);(5,8)(6,11)(7,10)(9,12)(13,16,18,28)(14,21,19,26)(15,24,23,35)(17,27,29,40)(20,32,30,37)(22,34,31,39)(25,38,36,43)(33,42,41,44);(13,17)(14,22)(15,25)(16,27)(18,29)(19,31)(20,33)(21,34)(23,36)(24,38)(26,39)(28,40)(30,41)(32,42)(35,43)(37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266731
maximal normal subgroup of	#72266731
maximal quotient of	#72266737, #72266767

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622078);