#7743487

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

8938 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236842, #72238148, #72238255
maximal normal subgroup of	#72236842, #72238148, #72238255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7650683);