#7938640

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

7211 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)(13,14,19,20)(15,21,16,18)(17,25,27,22)(23,28,24,26);(1,2)(3,4)(13,15,19,16)(14,18,20,21)(17,23,27,24)(22,28,25,26);(6,7)(10,11)(14,20)(15,16)(22,25)(23,24);(5,7,9,6)(8,11,12,10)(13,16,19,15)(14,21,20,18)(17,24,27,23)(22,26,25,28);(5,8,9,12)(6,10,7,11)(13,17,19,27)(14,22,20,25)(15,23,16,24)(18,28,21,26);(13,18,19,21)(14,15,20,16)(17,26,27,28)(22,24,25,23)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,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, 7845836);