#7658091

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	C₂×#3514	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7300 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)(7,12)(8,10)(9,11)(13,14,18,22)(15,19,20,16)(17,25,26,21)(23,24,28,27);(1,2,3,4)(13,15)(14,19)(16,22)(17,23)(18,20)(21,27)(24,25)(26,28);(5,7,9,10)(6,8,11,12)(13,16,18,19)(14,20,22,15)(17,24,26,27)(21,28,25,23);(13,17,18,26)(14,21,22,25)(15,23,20,28)(16,24,19,27);(5,6)(7,8)(9,11)(10,12);(5,8)(6,10)(7,11)(9,12)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7565287);