#7934104

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

7565 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)(7,11)(8,9)(10,12)(13,14)(15,24)(16,27)(17,28)(18,23)(19,20)(21,26)(22,25);(1,2,3,4)(13,15)(14,18)(16,25)(17,26)(19,23)(20,24)(21,22)(27,28);(6,9)(10,11)(14,20)(15,23)(16,26)(22,28);(5,6)(7,10)(8,9)(11,12)(13,16,19,26)(14,21,20,27)(15,25,23,17)(18,22,24,28);(5,7,8,12)(6,10,9,11)(13,17,19,25)(14,22,20,28)(15,26,23,16)(18,21,24,27);(13,18,19,24)(14,23,20,15)(16,22,26,28)(17,27,25,21)

Polycyclic presentation:

2^9|b^2=gi,d^2=i,e^2=h,f^2=i,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7841300);