#7693602

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

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

11369 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=i,d^2=h,e^2=i,f^2=i,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, 7600798);