#7897672

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

7178 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)(6,8)(10,11)(13,14,19,18)(15,23,25,20)(16,26,27,21)(17,28,24,22);(1,2,3,4)(13,15,19,25)(14,20,18,23)(16,24,27,17)(21,22,26,28);(5,6,9,8)(7,11,12,10)(13,16,19,27)(14,21,18,26)(15,17,25,24)(20,28,23,22);(5,7,9,12)(6,10,8,11)(13,16)(14,21)(15,24)(17,25)(18,26)(19,27)(20,22)(23,28);(5,8,9,6)(7,11,12,10)(13,17,19,24)(14,22,18,28)(15,16,25,27)(20,26,23,21);(13,18,19,14)(15,23,25,20)(16,26,27,21)(17,22,24,28)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=hi,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.

Relationships

maximal subgroup of	#72280781
maximal normal subgroup of	#72280781
maximal quotient of	#72280903, #72281298

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7804868);