#7736713

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

7410 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)(5,6,10,12)(7,11,8,9)(13,14,18,21)(15,19,22,16)(17,20,26,28)(23,27,25,24);(1,2)(3,4)(5,7,10,8)(6,11,12,9)(13,15)(14,19)(16,21)(17,25)(18,22)(20,24)(23,26)(27,28);(6,12)(7,8)(15,22)(16,19)(23,25)(24,27);(13,16,18,19)(14,15,21,22)(17,24,26,27)(20,23,28,25);(5,8,10,7)(6,11,12,9)(13,15)(14,19)(16,21)(17,23)(18,22)(20,27)(24,28)(25,26);(5,9,10,11)(6,7,12,8)(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,28)(22,25)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312841, #72312995, #72315197
maximal normal subgroup of	#72312841, #72312995, #72315197

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7643909);