#7697022

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

6986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(2,4)(5,6,10,12)(7,11,9,8)(13,14,18,16)(15,21,23,19)(17,20,25,24)(22,27,28,26);(1,2,3,4)(5,7,10,9)(6,11,12,8)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28);(6,12)(8,11)(13,15,18,23)(14,19,16,21)(17,22,25,28)(20,26,24,27);(5,8)(6,7)(9,12)(10,11)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28);(5,9,10,7)(6,8,12,11)(13,16,18,14)(15,21,23,19)(17,20,25,24)(22,26,28,27);(13,17)(14,20)(15,22)(16,24)(18,25)(19,26)(21,27)(23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7604218);