#7693609

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
prime radical	#39361	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7191 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7600805);