#7917259

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

7723 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)(5,6,8,11)(7,10,12,9)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12);(5,7)(6,9)(8,12)(10,11)(13,14,17,20)(15,18,23,21)(16,19,24,26)(22,25,28,27);(5,6)(7,10)(8,11)(9,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28);(5,6)(7,9)(8,11)(10,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(5,7)(6,10)(8,12)(9,11)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326930, #72330752
maximal normal subgroup of	#72326930, #72330752
maximal quotient of	#72326988, #72328389, #72329598, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7824455);