#7689882

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

8339 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)(6,10)(7,11)(13,14,18,21)(15,19,23,16)(17,20,26,28)(22,25,27,24);(1,2)(3,4)(5,6,9,10)(7,8,11,12)(13,14,18,21)(15,19,23,16)(17,20,26,28)(22,25,27,24);(5,7)(6,8)(9,11)(10,12)(13,15,18,23)(14,16,21,19)(17,22,26,27)(20,24,28,25);(5,8)(6,11)(7,10)(9,12)(13,15)(14,19)(16,21)(17,22)(18,23)(20,25)(24,28)(26,27);(13,16,18,19)(14,15,21,23)(17,25,26,24)(20,27,28,22);(13,17,18,26)(14,20,21,28)(15,22,23,27)(16,24,19,25)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304543, #72304587, #72304599
maximal normal subgroup of	#72304543, #72304587, #72304599

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7597078);