#8075293

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

7759 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)(7,10)(8,9)(11,12)(13,14)(15,17)(16,23)(18,20)(19,21)(22,25)(24,27)(26,28);(1,2)(3,4)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24)(25,28)(26,27);(6,9)(10,12)(13,16,19,17)(14,15,21,23)(18,25,26,24)(20,27,28,22);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,15,21,23)(18,24,26,25)(20,27,28,22);(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,21)(18,22)(19,23)(20,24)(25,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,18)(14,20)(15,22)(16,24)(17,25)(19,26)(21,28)(23,27)

Polycyclic presentation:

2^9|c^2=i,d^2=h,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	#72336567
maximal normal subgroup of	#72336567
maximal quotient of	#72336653

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982489);