#7715094

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

7572 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)(5,6,9,10)(7,11,12,8)(13,14,17,19)(15,18,21,26)(16,23,24,20)(22,28,27,25);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15,17,21)(14,18,19,26)(16,22,24,27)(20,25,23,28);(6,10)(8,11)(14,19)(15,21)(20,23)(22,27);(5,8)(6,7)(9,11)(10,12)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27);(5,7)(6,8)(9,12)(10,11)(13,14)(15,18)(16,20)(17,19)(21,26)(22,25)(23,24)(27,28);(13,16,17,24)(14,20,19,23)(15,22,21,27)(18,25,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238187, #72238191, #72238255
maximal normal subgroup of	#72238187, #72238191, #72238255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622290);