#7630602

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	#35565	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7364 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537798);