C₂²×#2868 #7633349

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

14562 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 20 points:

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

maximal subgroup of	C₂²×#36559, C₂²×#36644, C₂²×#36127, …
maximal normal subgroup of	C₂²×#36559, C₂²×#36644, C₂²×#36127, …

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540545);