#7690382

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

8897 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)(5,6,9,11)(7,8,12,10)(13,14,18,20)(15,19,23,25)(16,21,26,28)(17,22,27,24);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,24)(17,21)(18,23)(20,25)(22,26)(27,28);(2,4)(5,8,9,10)(6,7,11,12)(14,20)(15,23)(21,28)(22,24);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,19)(16,24)(17,21)(18,23)(20,25)(22,26)(27,28);(13,16)(14,21)(15,24)(17,19)(18,26)(20,28)(22,23)(25,27);(13,17)(14,22)(15,21)(16,25)(18,27)(19,26)(20,24)(23,28)

Polycyclic presentation:

2^9|a^2=gi,c^2=g,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	#72236842, #72238166, #72238250, …
maximal normal subgroup of	#72236842, #72238166, #72238250, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7597578);