#7657904

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

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

8146 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)(6,9)(7,11)(13,14)(15,24)(16,21)(17,20)(18,22)(19,23)(25,26)(27,28);(1,2)(3,4)(6,9)(10,12);(5,6,8,9)(7,10,11,12)(13,15,19,17)(14,20,23,24)(16,25,22,27)(18,26,21,28);(5,7)(6,10)(8,11)(9,12)(13,16,19,22)(14,21,23,18)(15,25,17,27)(20,28,24,26);(13,17,19,15)(14,20,23,24)(16,27,22,25)(18,26,21,28);(13,18,19,21)(14,22,23,16)(15,26,17,28)(20,27,24,25)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=hi,f^2=hi,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, 7565100);