#7985936

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

8457 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)(5,6,9,8)(7,11,12,10)(13,14,19,16)(15,20,18,22)(17,24,26,21)(23,28,25,27);(1,2)(3,4)(6,8)(10,11)(13,15,19,18)(14,20,16,22)(17,23,26,25)(21,27,24,28);(5,6,9,8)(7,10,12,11)(14,16)(20,22)(21,24)(27,28);(5,7)(6,10)(8,11)(9,12)(13,16,19,14)(15,22,18,20)(17,24,26,21)(23,28,25,27);(5,7)(6,11)(8,10)(9,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28);(5,8,9,6)(7,11,12,10)(13,18,19,15)(14,22,16,20)(17,25,26,23)(21,28,24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256671, #72256678, #72256682, …
maximal normal subgroup of	#72256671, #72256678, #72256682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7893132);