#8069699

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

7573 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)(5,6,8,9)(7,10,11,12)(13,14,18,20)(15,21,26,19)(16,22,27,25)(17,23,28,24);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,24)(17,25)(18,26)(20,21)(22,28)(23,27);(6,9)(7,11)(14,20)(17,28)(19,21)(22,25);(7,11)(10,12)(13,15)(14,21)(16,23)(17,25)(18,26)(19,20)(22,28)(24,27);(5,6)(7,10)(8,9)(11,12)(13,16)(14,22)(15,24)(17,21)(18,27)(19,28)(20,25)(23,26);(5,7)(6,10)(8,11)(9,12)(13,17)(14,23)(15,25)(16,21)(18,28)(19,27)(20,24)(22,26)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238192, #72238218, #72238245
maximal normal subgroup of	#72238192, #72238218, #72238245

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7976895);