#7670667

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

8500 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72305286
maximal normal subgroup of	#72305286
maximal quotient of	#72306253

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577863);