#8077211

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

7279 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)(6,9)(10,12)(13,14)(15,19)(16,21)(17,27)(18,20)(22,24)(23,28)(25,26);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,19,20,28)(16,24,26,17)(21,22,25,27);(6,9)(10,12)(14,20)(15,23)(17,24)(21,25);(6,9)(7,11)(13,16)(14,21)(15,17)(18,26)(19,27)(20,25)(22,28)(23,24);(5,6)(7,10)(8,9)(11,12)(13,17)(14,22)(15,16)(18,24)(19,25)(20,27)(21,28)(23,26);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,25)(17,22)(18,20)(21,26)(23,28)(24,27)

Polycyclic presentation:

2^9|b^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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	#72238198, #72238218, #72238242
maximal normal subgroup of	#72238198, #72238218, #72238242

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7984407);