#7684980

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

8395 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)(10,11)(13,14)(15,17)(16,21)(18,22)(19,20)(23,25)(24,26)(27,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,16)(14,17,20,21)(18,23,27,24)(22,25,28,26);(5,6)(7,11)(8,9)(10,12)(14,20)(15,16)(22,28)(23,24);(5,7)(6,10)(8,12)(9,11)(13,16,19,15)(14,21,20,17)(18,24,27,23)(22,26,28,25);(13,17,19,21)(14,15,20,16)(18,26,27,25)(22,24,28,23);(13,18,19,27)(14,22,20,28)(15,23,16,24)(17,25,21,26)

Polycyclic presentation:

2^9|b^2=gh,d^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, #72304599, #72304618
maximal normal subgroup of	#72304543, #72304599, #72304618

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592176);