#7677964

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

8006 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)(13,14,19,16)(15,23,25,20)(17,21,27,24)(18,26,28,22);(5,6,9,10)(7,8,11,12)(13,15,19,25)(14,20,16,23)(17,26,27,22)(18,24,28,21);(2,3)(5,7)(6,8)(9,11)(10,12)(13,14,19,16)(15,23,25,20)(17,24,27,21)(18,22,28,26);(1,3,4,2)(5,8)(6,11)(7,10)(9,12)(13,16,19,14)(15,23,25,20)(17,24,27,21)(18,26,28,22);(13,17,19,27)(14,21,16,24)(15,22,25,26)(18,20,28,23);(13,18)(14,22)(15,24)(16,26)(17,20)(19,28)(21,25)(23,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,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, 7585160);