#160357

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72266471, #72266643, #72267755
Fitting subgroup of	#72266471, #72266673, #72266674
maximal subgroup of	#72266471, #72266617, #72266622, …
maximal normal subgroup of	#72266471, #72266617, #72266622, …
maximal quotient of	#72266478, #72267759

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 67553);