#7680268

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

8213 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=gh,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7587464);