#7672667

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

8438 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)(5,6,8,10)(7,9,12,11)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,16)(15,19,21,22)(17,20,25,24)(23,26,27,28);(5,6)(7,11)(8,10)(9,12)(14,16)(15,21)(20,24)(23,27);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26);(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,e^2=i,f^2=i,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	#72302253, #72302972, #72302981
maximal normal subgroup of	#72302253, #72302972, #72302981

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579863);