#7744648

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

7580 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)(5,6,9,7)(8,11,12,10)(13,14,18,17)(15,19,22,20)(16,21,24,25)(23,26,28,27);(1,2,3,4)(13,15)(14,19)(16,23)(17,20)(18,22)(21,26)(24,28)(25,27);(6,7)(10,11)(15,22)(16,24)(19,20)(21,25);(5,7,9,6)(8,11,12,10)(13,15)(14,20)(16,23)(17,19)(18,22)(21,27)(24,28)(25,26);(5,8)(6,10)(7,11)(9,12)(13,16,18,24)(14,21,17,25)(15,23,22,28)(19,26,20,27);(13,17,18,14)(15,19,22,20)(16,25,24,21)(23,26,28,27)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,e^2=i,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	#72238158, #72238213, #72238232
maximal normal subgroup of	#72238158, #72238213, #72238232

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7651844);