#7678336

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

8178 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)(15,20)(16,17)(18,22)(19,25)(21,27)(23,24)(26,28);(5,6,9,10)(7,8,11,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,22)(21,23)(27,28);(2,3)(5,7)(6,8)(9,11)(10,12)(13,15,18,25)(14,19,22,20)(16,26,21,23)(17,24,27,28);(1,3,4,2)(5,8)(6,11)(7,10)(9,12)(13,15)(14,20)(16,23)(17,24)(18,25)(19,22)(21,26)(27,28);(13,16,18,21)(14,17,22,27)(15,23,25,26)(19,28,20,24);(13,17)(14,21)(15,24)(16,22)(18,27)(19,23)(20,26)(25,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=gh,e^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, 7585532);