#7985316

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

7420 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)(7,8)(9,11)(10,12)(13,14)(15,20)(16,25)(17,18)(19,21)(22,28)(23,27)(24,26);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15,19,16)(14,20,21,25)(17,23,28,26)(18,27,22,24);(5,8,10,9)(6,7,12,11)(14,21)(15,16)(17,28)(24,27);(5,7)(6,9)(8,12)(10,11)(13,16,19,15)(14,20,21,25)(17,26,28,23)(18,27,22,24);(5,7)(6,8)(9,12)(10,11)(13,17,19,28)(14,22,21,18)(15,23,16,26)(20,24,25,27);(5,9,10,8)(6,11,12,7)(13,18,19,22)(14,17,21,28)(15,24,16,27)(20,26,25,23)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315173, #72315185, #72315207
maximal normal subgroup of	#72315173, #72315185, #72315207

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892512);