#7989710

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
prime radical	#36121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7274 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=hi,e^2=hi,f^2=i,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	#72312247, #72312255, #72312370, …
maximal normal subgroup of	#72312247, #72312255, #72312370, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7896906);