#7679233

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	#34937	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6824 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=hi,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, 7586429);