#7806502

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

7084 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)(5,6,9,8)(7,11,12,10)(13,14,19,22)(15,20,18,16)(17,25,27,21)(23,24,26,28);(1,2,3,4)(5,7,9,12)(6,10,8,11)(13,15,19,18)(14,20,22,16)(17,23,27,26)(21,28,25,24);(7,12)(10,11)(13,16,19,20)(14,18,22,15)(17,24,27,28)(21,26,25,23);(5,6,9,8)(7,11,12,10)(13,16)(14,15)(17,24)(18,22)(19,20)(21,23)(25,26)(27,28);(5,8,9,6)(7,10,12,11)(13,17,19,27)(14,21,22,25)(15,23,18,26)(16,24,20,28);(5,7)(6,11)(8,10)(9,12)(13,18,19,15)(14,16,22,20)(17,26,27,23)(21,24,25,28)

Polycyclic presentation:

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

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, 7713698);