#7780005

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

7401 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)(13,14,19,18)(15,23,17,20)(16,22,26,21)(24,27,25,28);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,15,19,17)(14,20,18,23)(16,25,26,24)(21,28,22,27);(6,8)(10,12)(13,16,19,26)(14,21,18,22)(15,24,17,25)(20,27,23,28);(7,11)(10,12)(13,16)(14,22)(15,25)(17,24)(18,21)(19,26)(20,27)(23,28);(5,7)(6,10)(8,12)(9,11)(13,17,19,15)(14,23,18,20)(16,25,26,24)(21,28,22,27);(5,8,9,6)(7,12,11,10)(13,18,19,14)(15,23,17,20)(16,21,26,22)(24,27,25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7687201);