#7780281

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

8940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302988, #72303009
maximal normal subgroup of	#72302988, #72303009

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7687477);