#160295

order	512
nilpotency class	2
isoclinism	isoclinic class of #158815
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×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

2860 conjugacy classes of subgroups; only the 590 normal subgroups are shown.

normal subgroups only

This group has 2860 conjugacy classes of subgroups — too many to draw. Only its 590 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=efgh,d^2=e,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	#72267830
Fitting subgroup of	#72267830, #72267831, #72267871
maximal subgroup of	#72267820, #72267830, #72267860
maximal normal subgroup of	#72267820, #72267830, #72267860
maximal quotient of	#72267796

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 67491);