#7938621

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

8728 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)(13,14,19,21)(15,17,16,20)(18,22,27,28)(23,25,24,26);(1,2)(3,4)(13,15,19,16)(14,20,21,17)(18,23,27,24)(22,26,28,25);(6,7)(10,11)(14,21)(15,16)(22,28)(23,24);(5,7,9,6)(8,11,12,10)(13,16,19,15)(14,17,21,20)(18,24,27,23)(22,25,28,26);(5,8)(6,10)(7,11)(9,12)(13,17)(14,15)(16,21)(18,26)(19,20)(22,24)(23,28)(25,27);(13,18)(14,22)(15,23)(16,24)(17,25)(19,27)(20,26)(21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262026, #72262208
maximal normal subgroup of	#72262026, #72262208

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7845817);