#7681056

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
prime radical	#35518	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6905 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)(14,20)(18,22)(21,23)(25,27);(5,7,9,12)(6,10,8,11)(13,14)(15,23)(16,21)(17,18)(19,20)(22,28)(24,25)(26,27);(2,4)(5,6,9,8)(7,10,12,11)(13,15,19,16)(14,21,20,23)(17,26,28,24)(18,25,22,27);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,16,19,15)(14,21,20,23)(17,26,28,24)(18,27,22,25);(13,17)(14,22)(15,24)(16,26)(18,20)(19,28)(21,25)(23,27);(13,18,19,22)(14,17,20,28)(15,25,16,27)(21,26,23,24)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=g,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, 7588252);