#7715214

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

7576 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)(5,6,9,7)(8,11,12,10)(13,14,18,20)(15,19,16,21)(17,25,26,22)(23,28,24,27);(1,2,3,4)(13,15)(14,19)(16,18)(17,23)(20,21)(22,27)(24,26)(25,28);(6,7)(10,11)(14,20)(15,16)(22,25)(23,24);(5,7,9,6)(8,11,12,10)(13,16,18,15)(14,21,20,19)(17,24,26,23)(22,28,25,27);(5,8)(6,10)(7,11)(9,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,28)(25,26);(13,17,18,26)(14,22,20,25)(15,23,16,24)(19,27,21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238209, #72238213, #72238255
maximal normal subgroup of	#72238209, #72238213, #72238255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622410);