#7682768

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7589964);